I want to use a non-continuous line stencil for a border. How do you begin and end the sections? Do you backstitch a couple of stitches, or maybe tack a few stitches in place? How do you make sure the stitches don't come out when you clip the thread? Any help and hints are appreciated!

Study your design very well. This doesn't work with every design, but some non-continuous designs can be tricked into continuity. If you can't you can take a few stitches in place to secure your thread. You can then travel to the next area, secure before and after and then come back and clip all the loose ends.

If you are showing the quilt, you'll want to bury your threads, so stop, pull a long piece of thread, pull up the bobbin thread and cut, leaving about 3", then when you start the new pattern, pull the bobbin thread to the top, leave a tail about 3" long and start your pattern. When you're done, go to all those threads, tie them and use a needle to bury them into the quilt.
If you're not showing the quilt, just backstitch a couple of stitches, then lift the pressor foot to the next area, backstitch a couple of stitches and continue. When you're done, go back with scissors and snip away all the 'jump' stitches.
It's all in what you want to do with the quilt.
It's your quilt, so your rules apply! :)

Thanks, everyone, for the help! I'm not showing this quilt, in fact, it's my first time using stencil, so for this one I'll use the "take small stitches or backstitch a few" approach. But as I get more practice, I'm going to try the "bury the stitches". And tjradj, I like your "no rules" reminder :-)